% Mizar ND problem: t59_jordan2c,jordan2c,2578,21 fof(dh_c1_65__jordan2c,definition, ( ( m2_subset_1(c1_65__jordan2c,k1_numbers,k5_numbers) => ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & B = a_2_2_jordan2c(c1_65__jordan2c,A) ) => v2_connsp_1(B,k15_euclid(c1_65__jordan2c)) ) ) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m1_subset_1(D,k1_numbers) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(C)))) => ( ( r1_xreal_0(2,C) & E = a_2_2_jordan2c(C,D) ) => v2_connsp_1(E,k15_euclid(C)) ) ) ) ) ), introduced(definition,[new_symbol(c1_65__jordan2c),file(jordan2c,c1_65__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c1_65__jordan2c)]). fof(dh_c2_65__jordan2c,definition, ( ( m1_subset_1(c2_65__jordan2c,k1_numbers) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & A = a_2_2_jordan2c(c1_65__jordan2c,c2_65__jordan2c) ) => v2_connsp_1(A,k15_euclid(c1_65__jordan2c)) ) ) ) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & C = a_2_2_jordan2c(c1_65__jordan2c,B) ) => v2_connsp_1(C,k15_euclid(c1_65__jordan2c)) ) ) ) ), introduced(definition,[new_symbol(c2_65__jordan2c),file(jordan2c,c2_65__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c2_65__jordan2c)]). fof(dh_c3_65__jordan2c,definition, ( ( m1_subset_1(c3_65__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & c3_65__jordan2c = a_2_2_jordan2c(c1_65__jordan2c,c2_65__jordan2c) ) => v2_connsp_1(c3_65__jordan2c,k15_euclid(c1_65__jordan2c)) ) ) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & A = a_2_2_jordan2c(c1_65__jordan2c,c2_65__jordan2c) ) => v2_connsp_1(A,k15_euclid(c1_65__jordan2c)) ) ) ), introduced(definition,[new_symbol(c3_65__jordan2c),file(jordan2c,c3_65__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c3_65__jordan2c)]). fof(e1_65__jordan2c,assumption, ( r1_xreal_0(2,c1_65__jordan2c) & c3_65__jordan2c = a_2_2_jordan2c(c1_65__jordan2c,c2_65__jordan2c) ), introduced(assumption,[file(jordan2c,e1_65__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,e1_65__jordan2c)]). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), [interesting(0.9),axiom,file(finseq_2,k4_finseq_2)]). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(dt_u1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_funct_1(u1_metric_1(A)) & v1_funct_2(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) ) ), file(metric_1,u1_metric_1), [interesting(0.9),axiom,file(metric_1,u1_metric_1)]). fof(cc1_arytm_3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]). fof(cc1_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]). fof(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc2_arytm_3,theorem,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(rc1_arytm_3,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]). fof(rc1_finseq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]). fof(rc1_jordan2c,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v6_tbsp_1(B,A) ) ) ), file(jordan2c,rc1_jordan2c), [interesting(0.9),axiom,file(jordan2c,rc1_jordan2c)]). fof(rc2_finseq_1,theorem,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc2_finseq_1)]). fof(rc2_finset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), [interesting(0.9),axiom,file(finset_1,rc2_finset_1)]). fof(rc2_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v6_tbsp_1(B,A) ) ) ), file(tbsp_1,rc2_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,rc2_tbsp_1)]). fof(rc2_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]). fof(rc3_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc3_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v1_finset_1(B) & v6_tbsp_1(B,A) ) ) ), file(tbsp_1,rc3_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,rc3_tbsp_1)]). fof(rc3_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]). fof(rc4_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]). fof(rc4_xreal_0,theorem,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]). fof(rc8_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc8_finseq_1)]). fof(free_g1_metric_1,definition,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C,D] : ( g1_metric_1(A,B) = g1_metric_1(C,D) => ( A = C & B = D ) ) ) ), file(metric_1,g1_metric_1), [interesting(0.9),axiom,file(metric_1,g1_metric_1)]). fof(free_g1_pre_topc,definition,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ! [C,D] : ( g1_pre_topc(A,B) = g1_pre_topc(C,D) => ( A = C & B = D ) ) ) ), file(pre_topc,g1_pre_topc), [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]). fof(abstractness_v1_metric_1,theorem,( ! [A] : ( l1_metric_1(A) => ( v1_metric_1(A) => A = g1_metric_1(u1_struct_0(A),u1_metric_1(A)) ) ) ), file(metric_1,v1_metric_1), [interesting(0.9),axiom,file(metric_1,v1_metric_1)]). fof(existence_l1_metric_1,axiom,( ? [A] : l1_metric_1(A) ), file(metric_1,l1_metric_1), [interesting(0.9),axiom,file(metric_1,l1_metric_1)]). fof(dt_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( v1_metric_1(g1_metric_1(A,B)) & l1_metric_1(g1_metric_1(A,B)) ) ) ), file(metric_1,g1_metric_1), [interesting(0.9),axiom,file(metric_1,g1_metric_1)]). fof(dt_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( v1_pre_topc(g1_pre_topc(A,B)) & l1_pre_topc(g1_pre_topc(A,B)) ) ) ), file(pre_topc,g1_pre_topc), [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]). fof(dt_k13_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_funct_1(k13_euclid(A)) & v1_funct_2(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) & m2_relset_1(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) ) ), file(euclid,k13_euclid), [interesting(0.9),axiom,file(euclid,k13_euclid)]). fof(dt_k1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k1_euclid(A)) & m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ), file(euclid,k1_euclid), [interesting(0.9),axiom,file(euclid,k1_euclid)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(dt_l1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_struct_0(A) ) ), file(metric_1,l1_metric_1), [interesting(0.9),axiom,file(metric_1,l1_metric_1)]). fof(dt_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), [interesting(0.9),axiom,file(pre_topc,u1_pre_topc)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_finset_1,theorem,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), [interesting(0.9),axiom,file(finset_1,cc2_finset_1)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc2_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_finset_1(B) => v6_tbsp_1(B,A) ) ) ) ), file(tbsp_1,cc2_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,cc2_tbsp_1)]). fof(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). fof(cc3_arytm_3,theorem,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc3_arytm_3)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(fc14_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), [interesting(0.9),axiom,file(finset_1,fc14_finset_1)]). fof(fc2_finseq_1,theorem, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc2_finseq_1)]). fof(fc3_pcomps_1,theorem,( ! [A] : ( l1_metric_1(A) => ( v1_pre_topc(k5_pcomps_1(A)) & v2_pre_topc(k5_pcomps_1(A)) ) ) ), file(pcomps_1,fc3_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,fc3_pcomps_1)]). fof(fc4_pcomps_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ( ~ v3_struct_0(k5_pcomps_1(A)) & v1_pre_topc(k5_pcomps_1(A)) & v2_pre_topc(k5_pcomps_1(A)) ) ) ), file(pcomps_1,fc4_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,fc4_pcomps_1)]). fof(fc5_membered,theorem, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), [interesting(0.9),axiom,file(membered,fc5_membered)]). fof(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_finset_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]). fof(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). fof(rc1_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & v1_metric_1(A) ) ), file(metric_1,rc1_metric_1), [interesting(0.9),axiom,file(metric_1,rc1_metric_1)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(rc2_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) ) ), file(metric_1,rc2_metric_1), [interesting(0.9),axiom,file(metric_1,rc2_metric_1)]). fof(rc2_pcomps_1,theorem,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v3_compts_1(A) ) ), file(pcomps_1,rc2_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,rc2_pcomps_1)]). fof(rc3_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]). fof(rc3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]). fof(rc3_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) ) ), file(metric_1,rc3_metric_1), [interesting(0.9),axiom,file(metric_1,rc3_metric_1)]). fof(rc4_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), [interesting(0.9),axiom,file(finset_1,rc4_finset_1)]). fof(rc6_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(rc7_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]). fof(d1_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_euclid(A) = k4_finseq_2(A,k1_numbers) ) ), file(euclid,d1_euclid), [interesting(0.9),axiom,file(euclid,d1_euclid)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(abstractness_v1_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), [interesting(0.9),axiom,file(pre_topc,v1_pre_topc)]). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(existence_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), [interesting(0.9),axiom,file(pre_topc,m1_pre_topc)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k14_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) & l1_metric_1(k14_euclid(A)) ) ) ), file(euclid,k14_euclid), [interesting(0.9),axiom,file(euclid,k14_euclid)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k3_topmetr,axiom, ( v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & l1_pre_topc(k3_topmetr) ), file(topmetr,k3_topmetr), [interesting(0.9),axiom,file(topmetr,k3_topmetr)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_k5_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_pre_topc(k5_pcomps_1(A)) ) ), file(pcomps_1,k5_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,k5_pcomps_1)]). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(dt_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => l1_pre_topc(B) ) ) ), file(pre_topc,m1_pre_topc), [interesting(0.9),axiom,file(pre_topc,m1_pre_topc)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). fof(cc1_finset_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]). fof(cc1_funct_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]). fof(cc1_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => v2_pre_topc(B) ) ) ), file(pre_topc,cc1_pre_topc), [interesting(0.9),axiom,file(pre_topc,cc1_pre_topc)]). fof(cc1_relset_1,theorem,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]). fof(cc2_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(fc1_euclid,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k14_euclid(A)) & v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) ) ) ), file(euclid,fc1_euclid), [interesting(0.9),axiom,file(euclid,fc1_euclid)]). fof(fc1_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]). fof(fc2_euclid,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) ) ) ), file(euclid,fc2_euclid), [interesting(0.9),axiom,file(euclid,fc2_euclid)]). fof(fc2_membered,theorem, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), [interesting(0.9),axiom,file(membered,fc2_membered)]). fof(fc2_topmetr,theorem, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) ), file(topmetr,fc2_topmetr), [interesting(0.9),axiom,file(topmetr,fc2_topmetr)]). fof(fc2_topreal1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & v3_compts_1(k15_euclid(A)) ) ) ), file(topreal1,fc2_topreal1), [interesting(0.9),axiom,file(topreal1,fc2_topreal1)]). fof(fc3_pre_topc,theorem,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v3_struct_0(k3_pre_topc(A,B)) & v1_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc3_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc3_pre_topc)]). fof(fc3_topmetr,theorem,( ! [A,B,C] : ( ( v1_funct_1(B) & m1_relset_1(B,A,u1_struct_0(k3_topmetr)) ) => ( v1_xcmplx_0(k1_funct_1(B,C)) & v1_xreal_0(k1_funct_1(B,C)) ) ) ), file(topmetr,fc3_topmetr), [interesting(0.9),axiom,file(topmetr,fc3_topmetr)]). fof(fc4_subset_1,theorem,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), [interesting(0.9),axiom,file(subset_1,fc4_subset_1)]). fof(rc1_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]). fof(rc1_pre_topc,theorem,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc1_pre_topc)]). fof(rc1_subset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]). fof(rc2_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]). fof(rc2_pre_topc,theorem,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc2_pre_topc)]). fof(rc2_subset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]). fof(rc3_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) ) ) ), file(pre_topc,rc3_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc3_pre_topc)]). fof(rc3_struct_0,theorem,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]). fof(rc4_pre_topc,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & ~ v3_struct_0(B) & v1_pre_topc(B) ) ) ), file(pre_topc,rc4_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc4_pre_topc)]). fof(rc5_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc5_pre_topc)]). fof(rc5_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(d7_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k14_euclid(A) = g1_metric_1(k1_euclid(A),k13_euclid(A)) ) ), file(euclid,d7_euclid), [interesting(0.9),axiom,file(euclid,d7_euclid)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_k5_topmetr,definition,( k5_topmetr = k22_borsuk_1 ), file(topmetr,k5_topmetr), [interesting(0.9),axiom,file(topmetr,k5_topmetr)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_k15_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & l1_pre_topc(k15_euclid(A)) ) ) ), file(euclid,k15_euclid), [interesting(0.9),axiom,file(euclid,k15_euclid)]). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k22_borsuk_1,axiom,( l1_pre_topc(k22_borsuk_1) ), file(borsuk_1,k22_borsuk_1), [interesting(0.9),axiom,file(borsuk_1,k22_borsuk_1)]). fof(dt_k3_pre_topc,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & m1_pre_topc(k3_pre_topc(A,B),A) ) ) ), file(pre_topc,k3_pre_topc), [interesting(0.9),axiom,file(pre_topc,k3_pre_topc)]). fof(dt_k5_topmetr,axiom, ( v1_pre_topc(k5_topmetr) & m1_pre_topc(k5_topmetr,k3_topmetr) ), file(topmetr,k5_topmetr), [interesting(0.9),axiom,file(topmetr,k5_topmetr)]). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_struct_0)]). fof(dt_c1_65__jordan2c,assumption,( m2_subset_1(c1_65__jordan2c,k1_numbers,k5_numbers) ), introduced(assumption,[file(jordan2c,c1_65__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c1_65__jordan2c)]). fof(dt_c3_65__jordan2c,assumption,( m1_subset_1(c3_65__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) ), introduced(assumption,[file(jordan2c,c3_65__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c3_65__jordan2c)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(fc1_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), [interesting(0.9),axiom,file(subset_1,fc1_subset_1)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(d8_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k15_euclid(A) = k5_pcomps_1(k14_euclid(A)) ) ), file(euclid,d8_euclid), [interesting(0.9),axiom,file(euclid,d8_euclid)]). fof(de_c4_65__jordan2c,definition,( c4_65__jordan2c = c3_65__jordan2c ), introduced(definition,[new_symbol(c4_65__jordan2c),file(jordan2c,c4_65__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c4_65__jordan2c)]). fof(t2_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), [interesting(0.9),axiom,file(real,t2_real)]). fof(t3_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), [interesting(0.9),axiom,file(real,t3_real)]). fof(t5_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), [interesting(0.9),axiom,file(real,t5_real)]). fof(t6_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), [interesting(0.9),axiom,file(real,t6_real)]). fof(t7_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), [interesting(0.9),axiom,file(real,t7_real)]). fof(t8_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), [interesting(0.9),axiom,file(real,t8_real)]). fof(dt_k5_toprns_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k5_toprns_1(A,B),k1_numbers) ) ), file(toprns_1,k5_toprns_1), [interesting(0.9),axiom,file(toprns_1,k5_toprns_1)]). fof(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t4_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), [interesting(0.9),axiom,file(real,t4_real)]). fof(reflexivity_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(connectedness_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(dt_c2_65__jordan2c,assumption,( m1_subset_1(c2_65__jordan2c,k1_numbers) ), introduced(assumption,[file(jordan2c,c2_65__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c2_65__jordan2c)]). fof(rqLessOrEqual__r1_xreal_0__r2_r1,theorem,( ~ r1_xreal_0(2,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1)]). fof(rqLessOrEqual__r1_xreal_0__r2_r2,theorem,( r1_xreal_0(2,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.9),axiom,file(tarski,t2_tarski)]). fof(fraenkel_a_2_2_jordan2c,definition,( ! [A,B,C] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & m1_subset_1(C,k1_numbers) ) => ( r2_hidden(A,a_2_2_jordan2c(B,C)) <=> ? [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(B))) & A = D & ~ r1_xreal_0(k5_toprns_1(B,D),C) ) ) ) ), file(jordan2c,a_2_2_jordan2c), [interesting(0.9),axiom,file(jordan2c,a_2_2_jordan2c)]). fof(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc2_numerals,theorem, ( v2_xreal_0(2) & m2_subset_1(2,k1_numbers,k5_numbers) & m1_subset_1(2,k5_numbers) & m1_subset_1(2,k1_numbers) ), file(numerals,spc2_numerals), [interesting(0.9),axiom,file(numerals,spc2_numerals)]). fof(t2_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), [interesting(0.9),axiom,file(xreal_1,t2_xreal_1)]). fof(rqLessOrEqual__r1_xreal_0__r1_r2,theorem,( r1_xreal_0(1,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2)]). fof(rqLessOrEqual__r1_xreal_0__r1_r1,theorem,( r1_xreal_0(1,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1)]). fof(e2_65__jordan2c,plain,( r1_xreal_0(1,c1_65__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc14_finset_1,fc4_subset_1,rc1_jordan2c,rc2_finseq_1,rc2_tbsp_1,rc3_tbsp_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_l1_metric_1,dt_u1_pre_topc,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_tbsp_1,fc3_pcomps_1,fc4_pcomps_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_metric_1,rc2_finset_1,rc2_funct_1,rc2_metric_1,rc3_finseq_1,rc3_funct_1,rc3_metric_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_euclid,fc1_struct_0,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_pre_topc,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,d7_euclid,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xreal_0,rc2_subset_1,t1_real,t1_subset,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,d8_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,cc2_xreal_0,t2_tarski,fraenkel_a_2_2_jordan2c,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,e1_65__jordan2c,t2_xreal_1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r2_r1]), [interesting(0.8),file(jordan2c,e2_65__jordan2c),[file(jordan2c,e2_65__jordan2c)]]). fof(t58_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( r1_xreal_0(1,A) & a_2_2_jordan2c(A,B) = k1_xboole_0 ) ) ) ), file(jordan2c,t58_jordan2c), [interesting(0.9),axiom,file(jordan2c,t58_jordan2c)]). fof(e3_65__jordan2c,plain, ( ~ v1_xboole_0(c3_65__jordan2c) & m1_subset_1(c3_65__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc14_finset_1,fc4_subset_1,rc1_jordan2c,rc2_finseq_1,rc2_tbsp_1,rc3_tbsp_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_l1_metric_1,dt_u1_pre_topc,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_tbsp_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc3_pcomps_1,fc4_pcomps_1,rc1_arytm_3,rc1_metric_1,rc1_xreal_0,rc2_finset_1,rc2_metric_1,rc2_xreal_0,rc3_metric_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,d1_euclid,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_toprns_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_membered,cc20_membered,cc2_finset_1,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc1_struct_0,fc5_membered,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_pre_topc,rc2_funct_1,rc2_pcomps_1,rc2_pre_topc,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc4_funct_1,rc5_struct_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t1_real,t1_subset,t4_real,t4_subset,t5_subset,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc6_membered,cc9_membered,fc1_subset_1,fc2_euclid,fc2_finseq_1,fc2_membered,fc2_topreal1,fc6_membered,rc1_subset_1,rc2_subset_1,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,spc1_boole,spc2_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,t2_tarski,fraenkel_a_2_2_jordan2c,d8_euclid,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,e2_65__jordan2c,e1_65__jordan2c,t58_jordan2c,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2]), [interesting(0.8),file(jordan2c,e3_65__jordan2c),[file(jordan2c,e3_65__jordan2c)]]). fof(dt_c4_65__jordan2c,plain, ( ~ v1_xboole_0(c4_65__jordan2c) & m1_subset_1(c4_65__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_relset_1,cc2_arytm_3,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_tbsp_1,rc2_xreal_0,rc3_tbsp_1,rc3_xreal_0,rc4_xreal_0,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc1_finseq_1,cc1_xreal_0,cc2_funct_1,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc7_xreal_0,fc1_struct_0,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,rc1_finseq_1,rc1_funct_1,rc1_metric_1,rc1_xreal_0,rc2_funct_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finseq_1,rc3_funct_1,rc3_metric_1,rc3_struct_0,rc4_funct_1,rc5_struct_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,d1_euclid,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_xboole_0,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_finseq_1,fc2_membered,fc2_topreal1,fc6_membered,rc1_finset_1,rc1_membered,rc1_pre_topc,rc3_finset_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_65__jordan2c,dt_c3_65__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,fc1_subset_1,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d8_euclid,de_c4_65__jordan2c,e3_65__jordan2c]), [interesting(0.8),file(jordan2c,c4_65__jordan2c),[file(jordan2c,c4_65__jordan2c)]]). fof(fc4_pre_topc,theorem,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & v2_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc4_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc4_pre_topc)]). fof(fc5_borsuk_1,theorem, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) ), file(borsuk_1,fc5_borsuk_1), [interesting(0.9),axiom,file(borsuk_1,fc5_borsuk_1)]). fof(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(dh_c1_65_1__jordan2c,definition, ( ( m1_subset_1(c1_65_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) => ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) => ~ ( r2_hidden(c1_65_1__jordan2c,c4_65__jordan2c) & r2_hidden(A,c4_65__jordan2c) & c1_65_1__jordan2c != A & ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(B,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(B,0) & A = k1_funct_1(B,1) ) ) ) ) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(c1_65__jordan2c))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(c1_65__jordan2c))) => ~ ( r2_hidden(C,c4_65__jordan2c) & r2_hidden(D,c4_65__jordan2c) & C != D & ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(E,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & C = k1_funct_1(E,0) & D = k1_funct_1(E,1) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_65_1__jordan2c),file(jordan2c,c1_65_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c1_65_1__jordan2c)]). fof(dh_c2_65_1__jordan2c,definition, ( ( m1_subset_1(c2_65_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) => ~ ( r2_hidden(c1_65_1__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1__jordan2c,c4_65__jordan2c) & c1_65_1__jordan2c != c2_65_1__jordan2c & ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ) ) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) => ~ ( r2_hidden(c1_65_1__jordan2c,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & c1_65_1__jordan2c != B & ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(C,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(C,0) & B = k1_funct_1(C,1) ) ) ) ) ), introduced(definition,[new_symbol(c2_65_1__jordan2c),file(jordan2c,c2_65_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c2_65_1__jordan2c)]). fof(e1_65_1__jordan2c,assumption, ( r2_hidden(c1_65_1__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1__jordan2c,c4_65__jordan2c) & c1_65_1__jordan2c != c2_65_1__jordan2c ), introduced(assumption,[file(jordan2c,e1_65_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,e1_65_1__jordan2c)]). fof(e1_65_1_1_1__jordan2c,assumption,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( c1_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c2_65_1__jordan2c) & c2_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c1_65_1__jordan2c) ) ) ), introduced(assumption,[file(jordan2c,e1_65_1_1_1__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,e1_65_1_1_1__jordan2c)]). fof(dt_k1_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k1_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k1_topreal1), [interesting(0.9),axiom,file(topreal1,k1_topreal1)]). fof(fc1_topreal1,theorem,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => ~ v1_xboole_0(k1_topreal1(A,B,C)) ) ), file(topreal1,fc1_topreal1), [interesting(0.9),axiom,file(topreal1,fc1_topreal1)]). fof(commutativity_k3_topreal1,theorem,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k3_topreal1(A,B,C) = k3_topreal1(A,C,B) ) ), file(topreal1,k3_topreal1), [interesting(0.9),axiom,file(topreal1,k3_topreal1)]). fof(redefinition_k3_topreal1,definition,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k3_topreal1(A,B,C) = k1_topreal1(A,B,C) ) ), file(topreal1,k3_topreal1), [interesting(0.9),axiom,file(topreal1,k3_topreal1)]). fof(dt_k3_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k3_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k3_topreal1), [interesting(0.9),axiom,file(topreal1,k3_topreal1)]). fof(dt_c1_65_1__jordan2c,assumption,( m1_subset_1(c1_65_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), introduced(assumption,[file(jordan2c,c1_65_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c1_65_1__jordan2c)]). fof(dh_c1_65_1_1_1__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(A,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,B),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,B,c2_65_1__jordan2c),c4_65__jordan2c) ) ) => ( m1_subset_1(c1_65_1_1_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_1__jordan2c,c4_65__jordan2c) & r2_hidden(C,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_1__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_1__jordan2c,C),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,C,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ), introduced(definition,[new_symbol(c1_65_1_1_1__jordan2c),file(jordan2c,c1_65_1_1_1__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,c1_65_1_1_1__jordan2c)]). fof(dt_k18_euclid,axiom,( ! [A,B,C] : ( ( v1_xreal_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,u1_struct_0(k15_euclid(B))) ) => m1_subset_1(k18_euclid(A,B,C),u1_struct_0(k15_euclid(B))) ) ), file(euclid,k18_euclid), [interesting(0.9),axiom,file(euclid,k18_euclid)]). fof(dt_c2_65_1__jordan2c,assumption,( m1_subset_1(c2_65_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), introduced(assumption,[file(jordan2c,c2_65_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c2_65_1__jordan2c)]). fof(t49_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ~ ( C = a_2_2_jordan2c(A,B) & r2_hidden(D,C) & r2_hidden(E,C) & ! [F] : ( m1_subset_1(F,k1_numbers) => ( D != k18_euclid(F,A,E) & E != k18_euclid(F,A,D) ) ) & ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(F,C) & r2_hidden(G,C) & r1_tarski(k3_topreal1(A,D,F),C) & r1_tarski(k3_topreal1(A,F,G),C) & r1_tarski(k3_topreal1(A,G,E),C) ) ) ) ) ) ) ) ) ) ), file(jordan2c,t49_jordan2c), [interesting(0.9),axiom,file(jordan2c,t49_jordan2c)]). fof(e2_65_1_1_1__jordan2c,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(A,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,B),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,B,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_1__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_finseq_1,cc1_relset_1,cc2_funct_1,fc14_finset_1,fc4_subset_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_l1_metric_1,dt_u1_pre_topc,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_tbsp_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_metric_1,rc1_xreal_0,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_topreal1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_toprns_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc20_membered,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc1_struct_0,fc1_topreal1,fc5_membered,rc1_membered,rc1_pre_topc,rc1_subset_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t4_real,t5_subset,t6_boole,t8_boole,d7_euclid,commutativity_k3_topreal1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k3_topreal1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_topreal1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_65__jordan2c,dt_c1_65_1__jordan2c,dt_c2_65__jordan2c,dt_c2_65_1__jordan2c,dt_c3_65__jordan2c,dt_c4_65__jordan2c,de_c4_65__jordan2c,cc6_membered,cc9_membered,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t3_subset,t4_subset,t7_boole,t2_tarski,fraenkel_a_2_2_jordan2c,d8_euclid,spc2_numerals,spc2_boole,e1_65_1_1_1__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c,t49_jordan2c]), [interesting(0.35),file(jordan2c,e2_65_1_1_1__jordan2c),[file(jordan2c,e2_65_1_1_1__jordan2c)]]). fof(dt_c1_65_1_1_1__jordan2c,plain,( m1_subset_1(c1_65_1_1_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_1__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_1__jordan2c,e2_65_1_1_1__jordan2c]), [interesting(0.35),file(jordan2c,c1_65_1_1_1__jordan2c),[file(jordan2c,c1_65_1_1_1__jordan2c)]]). fof(dh_c2_65_1_1_1__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_1__jordan2c,c4_65__jordan2c) & r2_hidden(A,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_1__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_1__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,c2_65_1__jordan2c),c4_65__jordan2c) ) => ( m1_subset_1(c2_65_1_1_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_1__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_1__jordan2c,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_1__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_1__jordan2c,c2_65_1_1_1__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_1__jordan2c,c2_65_1__jordan2c),c4_65__jordan2c) ) ), introduced(definition,[new_symbol(c2_65_1_1_1__jordan2c),file(jordan2c,c2_65_1_1_1__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,c2_65_1_1_1__jordan2c)]). fof(dt_c2_65_1_1_1__jordan2c,plain,( m1_subset_1(c2_65_1_1_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_1__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_1__jordan2c,dh_c2_65_1_1_1__jordan2c,e2_65_1_1_1__jordan2c]), [interesting(0.35),file(jordan2c,c2_65_1_1_1__jordan2c),[file(jordan2c,c2_65_1_1_1__jordan2c)]]). fof(e3_65_1_1_1__jordan2c,plain, ( r2_hidden(c1_65_1_1_1__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_1__jordan2c,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_1__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_1__jordan2c,c2_65_1_1_1__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_1__jordan2c,c2_65_1__jordan2c),c4_65__jordan2c) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_1__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_1__jordan2c,dh_c2_65_1_1_1__jordan2c,e2_65_1_1_1__jordan2c]), [interesting(0.35),file(jordan2c,e3_65_1_1_1__jordan2c),[file(jordan2c,e3_65_1_1_1__jordan2c)]]). fof(t45_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(C,B) & r2_hidden(D,B) & r2_hidden(E,B) & r2_hidden(F,B) & r1_tarski(k3_topreal1(A,C,D),B) & r1_tarski(k3_topreal1(A,D,E),B) & r1_tarski(k3_topreal1(A,E,F),B) & ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) & m2_relset_1(G,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) ) => ~ ( v5_pre_topc(G,k5_topmetr,k3_pre_topc(k15_euclid(A),B)) & C = k1_funct_1(G,0) & F = k1_funct_1(G,1) ) ) ) ) ) ) ) ) ) ), file(jordan2c,t45_jordan2c), [interesting(0.9),axiom,file(jordan2c,t45_jordan2c)]). fof(e4_65_1_1_1__jordan2c,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_1__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[existence_m1_finseq_2,dt_k4_finseq_2,dt_m1_finseq_2,dt_u1_metric_1,cc1_finseq_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_jordan2c,rc2_finseq_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_l1_metric_1,dt_u1_pre_topc,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_tbsp_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_metric_1,rc1_xreal_0,rc2_finset_1,rc2_metric_1,rc3_finset_1,rc3_funct_1,rc3_metric_1,rc4_finset_1,rc6_finseq_1,rc7_finseq_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_relset_1,dt_k14_euclid,dt_k1_topreal1,dt_k22_borsuk_1,dt_k2_zfmisc_1,dt_k3_topmetr,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_relset_1,dt_c3_65__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_pre_topc,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,fc1_euclid,fc1_struct_0,fc1_topreal1,fc2_topmetr,fc3_pre_topc,fc3_topmetr,fc4_pre_topc,fc4_subset_1,fc5_borsuk_1,fc5_membered,rc1_funct_1,rc1_membered,rc1_pre_topc,rc1_subset_1,rc2_funct_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,d7_euclid,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k3_topreal1,redefinition_k5_numbers,redefinition_k5_topmetr,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k3_topreal1,dt_k5_numbers,dt_k5_topmetr,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_65__jordan2c,dt_c1_65_1__jordan2c,dt_c1_65_1_1_1__jordan2c,dt_c2_65_1__jordan2c,dt_c2_65_1_1_1__jordan2c,dt_c4_65__jordan2c,de_c4_65__jordan2c,cc6_membered,cc9_membered,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,t1_numerals,t1_subset,t3_subset,t4_subset,t7_boole,d8_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_65_1__jordan2c,e3_65_1_1_1__jordan2c,t45_jordan2c]), [interesting(0.35),file(jordan2c,e4_65_1_1_1__jordan2c),[file(jordan2c,e4_65_1_1_1__jordan2c)]]). fof(i2_65_1_1_1__jordan2c,theorem,( $true ), introduced(tautology,[file(jordan2c,i2_65_1_1_1__jordan2c)]), [interesting(0.35),trivial,file(jordan2c,i2_65_1_1_1__jordan2c)]). fof(i1_65_1_1_1__jordan2c,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_1__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[e4_65_1_1_1__jordan2c,i2_65_1_1_1__jordan2c]), [interesting(0.35),file(jordan2c,i1_65_1_1_1__jordan2c),[file(jordan2c,i1_65_1_1_1__jordan2c)]]). fof(i1_65_1_1__jordan2c,plain,( ~ ( ! [A] : ( m1_subset_1(A,k1_numbers) => ( c1_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c2_65_1__jordan2c) & c2_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c1_65_1__jordan2c) ) ) & ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c]),discharge_asm(discharge,[e1_65_1_1_1__jordan2c])],[e1_65_1_1_1__jordan2c,i1_65_1_1_1__jordan2c]), [interesting(0.5),file(jordan2c,i1_65_1_1__jordan2c),[file(jordan2c,i1_65_1_1__jordan2c)]]). fof(e1_65_1_1_2__jordan2c,assumption,( ~ ! [A] : ( m1_subset_1(A,k1_numbers) => ( c1_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c2_65_1__jordan2c) & c2_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c1_65_1__jordan2c) ) ) ), introduced(assumption,[file(jordan2c,e1_65_1_1_2__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,e1_65_1_1_2__jordan2c)]). fof(dh_c1_65_1_1_2__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(A,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & r2_hidden(C,c4_65__jordan2c) & r2_hidden(D,c4_65__jordan2c) & r2_hidden(E,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,B),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,B,C),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,C,D),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,D,E),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,E,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ) ) ) => ( m1_subset_1(c1_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [H] : ( m1_subset_1(H,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [I] : ( m1_subset_1(I,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(F,c4_65__jordan2c) & r2_hidden(G,c4_65__jordan2c) & r2_hidden(H,c4_65__jordan2c) & r2_hidden(I,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,F),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,F,G),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,G,H),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,H,I),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,I,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_65_1_1_2__jordan2c),file(jordan2c,c1_65_1_1_2__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,c1_65_1_1_2__jordan2c)]). fof(t56_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ~ ( r1_xreal_0(2,A) & C = a_2_2_jordan2c(A,B) & r2_hidden(D,C) & r2_hidden(E,C) & ~ ! [F] : ( m1_subset_1(F,k1_numbers) => ( D != k18_euclid(F,A,E) & E != k18_euclid(F,A,D) ) ) & ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(A))) => ! [H] : ( m1_subset_1(H,u1_struct_0(k15_euclid(A))) => ! [I] : ( m1_subset_1(I,u1_struct_0(k15_euclid(A))) => ! [J] : ( m1_subset_1(J,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(F,C) & r2_hidden(G,C) & r2_hidden(H,C) & r2_hidden(I,C) & r2_hidden(J,C) & r1_tarski(k3_topreal1(A,D,F),C) & r1_tarski(k3_topreal1(A,F,G),C) & r1_tarski(k3_topreal1(A,G,H),C) & r1_tarski(k3_topreal1(A,H,I),C) & r1_tarski(k3_topreal1(A,I,J),C) & r1_tarski(k3_topreal1(A,J,E),C) ) ) ) ) ) ) ) ) ) ) ) ) ), file(jordan2c,t56_jordan2c), [interesting(0.9),axiom,file(jordan2c,t56_jordan2c)]). fof(e2_65_1_1_2__jordan2c,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(A,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & r2_hidden(C,c4_65__jordan2c) & r2_hidden(D,c4_65__jordan2c) & r2_hidden(E,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,B),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,B,C),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,C,D),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,D,E),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,E,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_finseq_1,cc1_relset_1,cc2_funct_1,fc14_finset_1,fc4_subset_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_l1_metric_1,dt_u1_pre_topc,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_tbsp_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_metric_1,rc1_xreal_0,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_topreal1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_toprns_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc20_membered,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc1_struct_0,fc1_topreal1,fc5_membered,rc1_membered,rc1_pre_topc,rc1_subset_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t4_real,t5_subset,t6_boole,t8_boole,d7_euclid,commutativity_k3_topreal1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k3_topreal1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_topreal1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_65__jordan2c,dt_c1_65_1__jordan2c,dt_c2_65__jordan2c,dt_c2_65_1__jordan2c,dt_c3_65__jordan2c,dt_c4_65__jordan2c,de_c4_65__jordan2c,cc6_membered,cc9_membered,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,t1_subset,t3_subset,t4_subset,t7_boole,t2_tarski,fraenkel_a_2_2_jordan2c,d8_euclid,spc2_numerals,spc2_boole,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c,t56_jordan2c,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.35),file(jordan2c,e2_65_1_1_2__jordan2c),[file(jordan2c,e2_65_1_1_2__jordan2c)]]). fof(dt_c1_65_1_1_2__jordan2c,plain,( m1_subset_1(c1_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_2__jordan2c,e2_65_1_1_2__jordan2c]), [interesting(0.35),file(jordan2c,c1_65_1_1_2__jordan2c),[file(jordan2c,c1_65_1_1_2__jordan2c)]]). fof(dh_c2_65_1_1_2__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(A,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & r2_hidden(C,c4_65__jordan2c) & r2_hidden(D,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,B),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,B,C),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,C,D),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,D,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ) ) => ( m1_subset_1(c2_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(E,c4_65__jordan2c) & r2_hidden(F,c4_65__jordan2c) & r2_hidden(G,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,c2_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_2__jordan2c,E),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,E,F),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,F,G),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,G,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ) ) ), introduced(definition,[new_symbol(c2_65_1_1_2__jordan2c),file(jordan2c,c2_65_1_1_2__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,c2_65_1_1_2__jordan2c)]). fof(dt_c2_65_1_1_2__jordan2c,plain,( m1_subset_1(c2_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_2__jordan2c,dh_c2_65_1_1_2__jordan2c,e2_65_1_1_2__jordan2c]), [interesting(0.35),file(jordan2c,c2_65_1_1_2__jordan2c),[file(jordan2c,c2_65_1_1_2__jordan2c)]]). fof(dh_c3_65_1_1_2__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(A,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & r2_hidden(C,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,c2_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_2__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,B),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,B,C),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,C,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ) => ( m1_subset_1(c3_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c3_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(D,c4_65__jordan2c) & r2_hidden(E,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,c2_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_2__jordan2c,c3_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c3_65_1_1_2__jordan2c,D),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,D,E),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,E,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ) ), introduced(definition,[new_symbol(c3_65_1_1_2__jordan2c),file(jordan2c,c3_65_1_1_2__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,c3_65_1_1_2__jordan2c)]). fof(dt_c3_65_1_1_2__jordan2c,plain,( m1_subset_1(c3_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_2__jordan2c,dh_c2_65_1_1_2__jordan2c,dh_c3_65_1_1_2__jordan2c,e2_65_1_1_2__jordan2c]), [interesting(0.35),file(jordan2c,c3_65_1_1_2__jordan2c),[file(jordan2c,c3_65_1_1_2__jordan2c)]]). fof(dh_c4_65_1_1_2__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c3_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(A,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,c2_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_2__jordan2c,c3_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c3_65_1_1_2__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,B),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,B,c2_65_1__jordan2c),c4_65__jordan2c) ) ) => ( m1_subset_1(c4_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) & ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c3_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c4_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(C,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,c2_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_2__jordan2c,c3_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c3_65_1_1_2__jordan2c,c4_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c4_65_1_1_2__jordan2c,C),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,C,c2_65_1__jordan2c),c4_65__jordan2c) ) ) ), introduced(definition,[new_symbol(c4_65_1_1_2__jordan2c),file(jordan2c,c4_65_1_1_2__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,c4_65_1_1_2__jordan2c)]). fof(dt_c4_65_1_1_2__jordan2c,plain,( m1_subset_1(c4_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_2__jordan2c,dh_c2_65_1_1_2__jordan2c,dh_c3_65_1_1_2__jordan2c,dh_c4_65_1_1_2__jordan2c,e2_65_1_1_2__jordan2c]), [interesting(0.35),file(jordan2c,c4_65_1_1_2__jordan2c),[file(jordan2c,c4_65_1_1_2__jordan2c)]]). fof(dh_c5_65_1_1_2__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c3_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c4_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(A,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,c2_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_2__jordan2c,c3_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c3_65_1_1_2__jordan2c,c4_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c4_65_1_1_2__jordan2c,A),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,A,c2_65_1__jordan2c),c4_65__jordan2c) ) => ( m1_subset_1(c5_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) & r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c3_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c4_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c5_65_1_1_2__jordan2c,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,c2_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_2__jordan2c,c3_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c3_65_1_1_2__jordan2c,c4_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c4_65_1_1_2__jordan2c,c5_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c5_65_1_1_2__jordan2c,c2_65_1__jordan2c),c4_65__jordan2c) ) ), introduced(definition,[new_symbol(c5_65_1_1_2__jordan2c),file(jordan2c,c5_65_1_1_2__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,c5_65_1_1_2__jordan2c)]). fof(dt_c5_65_1_1_2__jordan2c,plain,( m1_subset_1(c5_65_1_1_2__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_2__jordan2c,dh_c2_65_1_1_2__jordan2c,dh_c3_65_1_1_2__jordan2c,dh_c4_65_1_1_2__jordan2c,dh_c5_65_1_1_2__jordan2c,e2_65_1_1_2__jordan2c]), [interesting(0.35),file(jordan2c,c5_65_1_1_2__jordan2c),[file(jordan2c,c5_65_1_1_2__jordan2c)]]). fof(e3_65_1_1_2__jordan2c,plain, ( r2_hidden(c1_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c3_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c4_65_1_1_2__jordan2c,c4_65__jordan2c) & r2_hidden(c5_65_1_1_2__jordan2c,c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1__jordan2c,c1_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c1_65_1_1_2__jordan2c,c2_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c2_65_1_1_2__jordan2c,c3_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c3_65_1_1_2__jordan2c,c4_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c4_65_1_1_2__jordan2c,c5_65_1_1_2__jordan2c),c4_65__jordan2c) & r1_tarski(k3_topreal1(c1_65__jordan2c,c5_65_1_1_2__jordan2c,c2_65_1__jordan2c),c4_65__jordan2c) ), inference(consider,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[dh_c1_65_1_1_2__jordan2c,dh_c2_65_1_1_2__jordan2c,dh_c3_65_1_1_2__jordan2c,dh_c4_65_1_1_2__jordan2c,dh_c5_65_1_1_2__jordan2c,e2_65_1_1_2__jordan2c]), [interesting(0.35),file(jordan2c,e3_65_1_1_2__jordan2c),[file(jordan2c,e3_65_1_1_2__jordan2c)]]). fof(t46_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(A))) => ! [H] : ( m1_subset_1(H,u1_struct_0(k15_euclid(A))) => ! [I] : ( m1_subset_1(I,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(C,B) & r2_hidden(D,B) & r2_hidden(E,B) & r2_hidden(F,B) & r2_hidden(G,B) & r2_hidden(H,B) & r2_hidden(I,B) & r1_tarski(k3_topreal1(A,C,D),B) & r1_tarski(k3_topreal1(A,D,E),B) & r1_tarski(k3_topreal1(A,E,F),B) & r1_tarski(k3_topreal1(A,F,G),B) & r1_tarski(k3_topreal1(A,G,H),B) & r1_tarski(k3_topreal1(A,H,I),B) & ! [J] : ( ( v1_funct_1(J) & v1_funct_2(J,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) & m2_relset_1(J,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) ) => ~ ( v5_pre_topc(J,k5_topmetr,k3_pre_topc(k15_euclid(A),B)) & C = k1_funct_1(J,0) & I = k1_funct_1(J,1) ) ) ) ) ) ) ) ) ) ) ) ) ), file(jordan2c,t46_jordan2c), [interesting(0.9),axiom,file(jordan2c,t46_jordan2c)]). fof(e4_65_1_1_2__jordan2c,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[existence_m1_finseq_2,dt_k4_finseq_2,dt_m1_finseq_2,dt_u1_metric_1,cc1_finseq_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_jordan2c,rc2_finseq_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_l1_metric_1,dt_u1_pre_topc,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_tbsp_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_metric_1,rc1_xreal_0,rc2_finset_1,rc2_metric_1,rc3_finset_1,rc3_funct_1,rc3_metric_1,rc4_finset_1,rc6_finseq_1,rc7_finseq_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_relset_1,dt_k14_euclid,dt_k1_topreal1,dt_k22_borsuk_1,dt_k2_zfmisc_1,dt_k3_topmetr,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_relset_1,dt_c3_65__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_pre_topc,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,fc1_euclid,fc1_struct_0,fc1_topreal1,fc2_topmetr,fc3_pre_topc,fc3_topmetr,fc4_pre_topc,fc4_subset_1,fc5_borsuk_1,fc5_membered,rc1_funct_1,rc1_membered,rc1_pre_topc,rc1_subset_1,rc2_funct_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,d7_euclid,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k3_topreal1,redefinition_k5_numbers,redefinition_k5_topmetr,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k3_topreal1,dt_k5_numbers,dt_k5_topmetr,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_65__jordan2c,dt_c1_65_1__jordan2c,dt_c1_65_1_1_2__jordan2c,dt_c2_65_1__jordan2c,dt_c2_65_1_1_2__jordan2c,dt_c3_65_1_1_2__jordan2c,dt_c4_65__jordan2c,dt_c4_65_1_1_2__jordan2c,dt_c5_65_1_1_2__jordan2c,de_c4_65__jordan2c,cc6_membered,cc9_membered,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,t1_numerals,t1_subset,t3_subset,t4_subset,t7_boole,d8_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_65_1__jordan2c,e3_65_1_1_2__jordan2c,t46_jordan2c]), [interesting(0.35),file(jordan2c,e4_65_1_1_2__jordan2c),[file(jordan2c,e4_65_1_1_2__jordan2c)]]). fof(i2_65_1_1_2__jordan2c,theorem,( $true ), introduced(tautology,[file(jordan2c,i2_65_1_1_2__jordan2c)]), [interesting(0.35),trivial,file(jordan2c,i2_65_1_1_2__jordan2c)]). fof(i1_65_1_1_2__jordan2c,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65_1_1_2__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c])],[e4_65_1_1_2__jordan2c,i2_65_1_1_2__jordan2c]), [interesting(0.35),file(jordan2c,i1_65_1_1_2__jordan2c),[file(jordan2c,i1_65_1_1_2__jordan2c)]]). fof(i2_65_1_1__jordan2c,plain,( ~ ( ~ ! [A] : ( m1_subset_1(A,k1_numbers) => ( c1_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c2_65_1__jordan2c) & c2_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c1_65_1__jordan2c) ) ) & ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,dt_c1_65__jordan2c,dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c]),discharge_asm(discharge,[e1_65_1_1_2__jordan2c])],[e1_65_1_1_2__jordan2c,i1_65_1_1_2__jordan2c]), [interesting(0.5),file(jordan2c,i2_65_1_1__jordan2c),[file(jordan2c,i2_65_1_1__jordan2c)]]). fof(e1_65_1_1__jordan2c,plain,( ~ ( ~ ! [A] : ( m1_subset_1(A,k1_numbers) => ( c1_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c2_65_1__jordan2c) & c2_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c1_65_1__jordan2c) ) ) & ! [A] : ( m1_subset_1(A,k1_numbers) => ( c1_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c2_65_1__jordan2c) & c2_65_1__jordan2c != k18_euclid(A,c1_65__jordan2c,c1_65_1__jordan2c) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_65__jordan2c,dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc14_finset_1,fc4_subset_1,rc1_jordan2c,rc2_finseq_1,rc2_tbsp_1,rc3_tbsp_1,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_l1_metric_1,dt_u1_pre_topc,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_tbsp_1,fc3_pcomps_1,fc4_pcomps_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_metric_1,rc2_finset_1,rc2_funct_1,rc2_metric_1,rc3_finseq_1,rc3_funct_1,rc3_metric_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k5_numbers,dt_m2_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc4_membered,cc7_xreal_0,fc2_euclid,fc2_topreal1,t2_subset,t6_boole,t7_boole,t8_boole,d8_euclid,existence_m1_subset_1,dt_k18_euclid,dt_k1_numbers,dt_m1_subset_1,dt_c1_65__jordan2c,dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,fc2_membered]), [interesting(0.5),file(jordan2c,e1_65_1_1__jordan2c),[file(jordan2c,e1_65_1_1__jordan2c)]]). fof(i2_65_1__jordan2c,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ), inference(percases,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c,e1_65_1__jordan2c,dt_c1_65__jordan2c,dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c])],[i1_65_1_1__jordan2c,i2_65_1_1__jordan2c,e1_65_1_1__jordan2c]), [interesting(0.65),file(jordan2c,i2_65_1__jordan2c),[file(jordan2c,i2_65_1__jordan2c)]]). fof(i1_65_1__jordan2c,plain,( ~ ( r2_hidden(c1_65_1__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1__jordan2c,c4_65__jordan2c) & c1_65_1__jordan2c != c2_65_1__jordan2c & ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c,dt_c1_65__jordan2c,dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c]),discharge_asm(discharge,[e1_65_1__jordan2c])],[e1_65_1__jordan2c,i2_65_1__jordan2c]), [interesting(0.65),file(jordan2c,i1_65_1__jordan2c),[file(jordan2c,i1_65_1__jordan2c)]]). fof(i1_65_1_tmp__jordan2c,plain, ( ( m1_subset_1(c1_65_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) & m1_subset_1(c2_65_1__jordan2c,u1_struct_0(k15_euclid(c1_65__jordan2c))) ) => ~ ( r2_hidden(c1_65_1__jordan2c,c4_65__jordan2c) & r2_hidden(c2_65_1__jordan2c,c4_65__jordan2c) & c1_65_1__jordan2c != c2_65_1__jordan2c & ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(A,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & c1_65_1__jordan2c = k1_funct_1(A,0) & c2_65_1__jordan2c = k1_funct_1(A,1) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c,dt_c1_65__jordan2c]),discharge_asm(discharge,[dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c])],[dt_c1_65_1__jordan2c,dt_c2_65_1__jordan2c,i1_65_1__jordan2c]), [interesting(0.8),e4_65__jordan2c]). fof(e4_65__jordan2c,plain,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(c1_65__jordan2c))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(c1_65__jordan2c))) => ~ ( r2_hidden(A,c4_65__jordan2c) & r2_hidden(B,c4_65__jordan2c) & A != B & ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c))) ) => ~ ( v5_pre_topc(C,k5_topmetr,k3_pre_topc(k15_euclid(c1_65__jordan2c),c4_65__jordan2c)) & A = k1_funct_1(C,0) & B = k1_funct_1(C,1) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c,dt_c1_65__jordan2c])],[i1_65_1_tmp__jordan2c,dh_c1_65_1__jordan2c,dh_c2_65_1__jordan2c]), [interesting(0.8),file(jordan2c,e4_65__jordan2c),[file(jordan2c,e4_65__jordan2c)]]). fof(t5_jordan1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ~ ( r2_hidden(C,B) & r2_hidden(D,B) & C != D & ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(A,B))) & m2_relset_1(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(A,B))) ) => ~ ( v5_pre_topc(E,k22_borsuk_1,k3_pre_topc(A,B)) & C = k1_funct_1(E,0) & D = k1_funct_1(E,1) ) ) ) ) ) => v2_connsp_1(B,A) ) ) ) ), file(jordan1,t5_jordan1), [interesting(0.9),axiom,file(jordan1,t5_jordan1)]). fof(e5_65__jordan2c,plain,( v2_connsp_1(c3_65__jordan2c,k15_euclid(c1_65__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c,dt_c1_65__jordan2c])],[existence_m1_finseq_2,dt_k4_finseq_2,dt_m1_finseq_2,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_finseq_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc3_finset_1,rc3_funct_1,rc3_metric_1,rc4_finset_1,rc6_finseq_1,rc7_finseq_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k2_zfmisc_1,dt_k3_topmetr,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_relset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc1_relset_1,cc2_funct_1,cc6_membered,cc9_membered,fc1_euclid,fc1_struct_0,fc2_euclid,fc2_membered,fc2_topmetr,fc2_topreal1,fc3_pre_topc,fc3_topmetr,fc4_subset_1,rc1_funct_1,rc1_pre_topc,rc1_subset_1,rc2_funct_1,rc2_pre_topc,rc2_subset_1,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t1_numerals,t2_subset,t5_subset,t6_boole,t8_boole,d7_euclid,antisymmetry_r2_hidden,existence_l1_pre_topc,existence_m1_subset_1,existence_m2_relset_1,redefinition_k5_topmetr,redefinition_m2_relset_1,dt_k15_euclid,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k22_borsuk_1,dt_k3_pre_topc,dt_k5_topmetr,dt_l1_pre_topc,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_65__jordan2c,dt_c3_65__jordan2c,dt_c4_65__jordan2c,de_c4_65__jordan2c,fc1_subset_1,fc4_pre_topc,fc5_borsuk_1,t1_subset,t3_subset,t4_subset,t7_boole,d8_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e4_65__jordan2c,t5_jordan1]), [interesting(0.8),file(jordan2c,e5_65__jordan2c),[file(jordan2c,e5_65__jordan2c)]]). fof(i5_65__jordan2c,theorem,( $true ), introduced(tautology,[file(jordan2c,i5_65__jordan2c)]), [interesting(0.8),trivial,file(jordan2c,i5_65__jordan2c)]). fof(i4_65__jordan2c,plain,( v2_connsp_1(c3_65__jordan2c,k15_euclid(c1_65__jordan2c)) ), inference(conclusion,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c3_65__jordan2c,e1_65__jordan2c,dt_c1_65__jordan2c])],[e5_65__jordan2c,i5_65__jordan2c]), [interesting(0.8),file(jordan2c,i4_65__jordan2c),[file(jordan2c,i4_65__jordan2c)]]). fof(i3_65__jordan2c,plain, ( ( r1_xreal_0(2,c1_65__jordan2c) & c3_65__jordan2c = a_2_2_jordan2c(c1_65__jordan2c,c2_65__jordan2c) ) => v2_connsp_1(c3_65__jordan2c,k15_euclid(c1_65__jordan2c)) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c3_65__jordan2c,dt_c1_65__jordan2c]),discharge_asm(discharge,[e1_65__jordan2c])],[e1_65__jordan2c,i4_65__jordan2c]), [interesting(0.8),file(jordan2c,i3_65__jordan2c),[file(jordan2c,i3_65__jordan2c)]]). fof(i3_65_tmp__jordan2c,plain, ( m1_subset_1(c3_65__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & c3_65__jordan2c = a_2_2_jordan2c(c1_65__jordan2c,c2_65__jordan2c) ) => v2_connsp_1(c3_65__jordan2c,k15_euclid(c1_65__jordan2c)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c1_65__jordan2c]),discharge_asm(discharge,[dt_c3_65__jordan2c])],[dt_c3_65__jordan2c,i3_65__jordan2c]), [interesting(0.8),i2_65__jordan2c]). fof(i2_65__jordan2c,plain,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & A = a_2_2_jordan2c(c1_65__jordan2c,c2_65__jordan2c) ) => v2_connsp_1(A,k15_euclid(c1_65__jordan2c)) ) ) ), inference(let,[status(thm),assumptions([dt_c2_65__jordan2c,dt_c1_65__jordan2c])],[i3_65_tmp__jordan2c,dh_c3_65__jordan2c]), [interesting(0.8),file(jordan2c,i2_65__jordan2c),[file(jordan2c,i2_65__jordan2c)]]). fof(i2_65_tmp__jordan2c,plain, ( m1_subset_1(c2_65__jordan2c,k1_numbers) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & A = a_2_2_jordan2c(c1_65__jordan2c,c2_65__jordan2c) ) => v2_connsp_1(A,k15_euclid(c1_65__jordan2c)) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_65__jordan2c]),discharge_asm(discharge,[dt_c2_65__jordan2c])],[dt_c2_65__jordan2c,i2_65__jordan2c]), [interesting(0.8),i1_65__jordan2c]). fof(i1_65__jordan2c,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & B = a_2_2_jordan2c(c1_65__jordan2c,A) ) => v2_connsp_1(B,k15_euclid(c1_65__jordan2c)) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_65__jordan2c])],[i2_65_tmp__jordan2c,dh_c2_65__jordan2c]), [interesting(0.8),file(jordan2c,i1_65__jordan2c),[file(jordan2c,i1_65__jordan2c)]]). fof(i1_65_tmp__jordan2c,plain, ( m2_subset_1(c1_65__jordan2c,k1_numbers,k5_numbers) => ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_65__jordan2c)))) => ( ( r1_xreal_0(2,c1_65__jordan2c) & B = a_2_2_jordan2c(c1_65__jordan2c,A) ) => v2_connsp_1(B,k15_euclid(c1_65__jordan2c)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_65__jordan2c])],[dt_c1_65__jordan2c,i1_65__jordan2c]), [interesting(1),t59_jordan2c]). fof(t59_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( r1_xreal_0(2,A) & C = a_2_2_jordan2c(A,B) ) => v2_connsp_1(C,k15_euclid(A)) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_65_tmp__jordan2c,dh_c1_65__jordan2c]), [interesting(1),file(jordan2c,t59_jordan2c),[file(jordan2c,t59_jordan2c)]]).